The asymptotic behaviour of the counting functions of $\varOmega $-sets in arithmetical semigroups
Volume 163 / 2014
Abstract
We consider an axiomatically-defined class of arithmetical semigroups that we call simple $L$-semigroups. This class includes all generalized Hilbert semigroups, in particular the semigroup of non-zero integers in any algebraic number field. We show, for all positive integers $k$, that the counting function of the set of elements with at most $k$ distinct factorization lengths in such a semigroup has oscillations of logarithmic frequency and size $\sqrt {x}(\log x)^{-M}$ for some $M>0$. More generally, we show a result on oscillations of counting functions of a family of subsets of simple $L$-semigroups. As another application we obtain similar results for the set of positive (rational) integers and the set of ideals in a ring of algebraic integers without non-trivial divisors in a given arithmetic progression.
